Statistical parametric mapping for event-related potentials (II): a hierarchical temporal model

Transcription

1 Statistical parametric mapping for event-related potentials (II): a hierarchical temporal model Stefan J. Kiebel* and Karl J. Friston Functional Imaging Laboratory, Institute of Neurology, Wellcome Department of Imaging Neuroscience, London WC1N 3BG, UK Received 16 October 2003; revised 7 February 2004; accepted 12 February NeuroImage 22 (2004) In this paper, we describe a temporal model for event-related potentials (ERP) in the context of statistical parametric mapping (SPM). In brief, we project channel data onto a two-dimensional scalp surface or into three-dimensional brain space using some appropriate inverse solution. We then treat the spatiotemporal data in a mass-univariate fashion. This implicitly factorises the model into spatial and temporal components. The key contribution of this paper is the use of observation models that afford an explicit distinction between observation error and variation in the expression of ERPs. This distinction is created by employing a two-level hierarchical model, in which the first level models the ERP effects within-subject and trial type, while the second models differences in ERP expression among trial types and subjects. By bringing the analysis of ERP data into a classical hierarchical (i.e., mixed effects) framework, many apparently disparate approaches (e.g., conventional P300 analyses and timefrequency analyses of stimulus-locked oscillations) can be reconciled within the same estimation and inference procedure. Inference proceeds in the normal way using t or F statistics to test for effects that are localised in peristimulus time or in some time frequency window. The use of F statistics is an important generalisation of classical approaches, because it allows one to test for effects that lie in a multidimensional subspace (i.e., of unknown but constrained form). We describe the analysis procedures, the underlying theory and compare its performance to established techniques. D 2004 Elsevier Inc. All rights reserved. Keywords: Statistical parametric mapping; Event-related potentials; Temporal model Introduction Electroencephalography (EEG) measures voltage changes on the scalp induced by underlying neuronal activity. The EEG is thought to be caused by postsynaptic potential changes in cortical pyramidal neurons (Lopes da Silva and van Rotterdam, 1982). An important * Corresponding author. Functional Imaging Laboratory, Institute of Neurology, Wellcome Department of Imaging Neuroscience, 12 Queen Square, London WC1N 3BG, UK. Fax: address: skiebel@fil.ion.ucl.ac.uk (S.J. Kiebel). Available online on ScienceDirect ( application of EEG is to assess differences among responses to different stimuli. Traditionally, one distinguishes between some stimulus-locked component and a residual (error) process. The stimulus-locked component is called the event-related potential (ERP). Often, because of a low signal-to-noise ratio, one measures several responses to estimate the average ERP, i.e., the common component of all ERPs with respect to a given stimulus type. In cognitive neuroscience, the analysis of multisubject-averaged ERP data is well established. For example, guidelines for recording standards and publication criteria are described in Picton et al. (2000). In an ERP study, one analyses ERPs of several subjects under several trial types (conditions). The analysis proceeds, at each channel, by estimating a contrast or linear compound of the ERP for each subject. Typically, these linear combinations are averages over peristimulus time windows. The statistical test of differences among these subject-specific, trial type-specific contrasts is based on an analysis of variance (ANOVA) with appropriate corrections for nonsphericity. This or a similar procedure represents the conventional ERP analysis. Another, more recent, analysis procedure is based on time frequency decomposition. Here, the hypotheses relate to power in a specific frequency range within a peristimulus time window. This kind of analysis is often used on single trial data to characterise induced responses (non-time-locked), but can also be applied to averaged ERP data (Tallon-Baudry et al., 1998). The approach usually involves a continuous or discrete wavelet transform. The wavelet transform has also been used to reduce the ERP to one or a few wavelet parameters that capture, parsimoniously, the differences among trial types or groups of subjects. Some authors have used wavelet parameter estimates in a descriptive fashion (Thakor et al., 1993) or have derived statistics as the basis of inferences about ERP differences (Basar et al., 1999; Trejo and Shensa, 1999). Both the conventional and time frequency analyses have been used to detect and make classical inferences about differences, between either trial types or groups of subjects. In this paper, we show that both analyses can proceed within a unified statistical framework using the same model. We propose a two-level hierarchical linear model as a general observation model for ERP data. In this model, conventional and time frequency hypotheses can be tested using single- or multidimensional contrasts. The ensuing statistics are either t-or F-distributed, where the degrees of freedom /$ - see front matter D 2004 Elsevier Inc. All rights reserved. doi: /j.neuroimage

2 504 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) are adjusted for nonsphericity. As an alternative to t- or F-statistics based on ordinary least squares, one can use the nonsphericity estimate to whiten the data and derive maximum-likelihood estimators. Inferences can be made at both levels of the model, resulting either in a fixed effects or random effects analysis. In addition to covering established characterisations, we will give examples of other useful contrasts that arise naturally from our framework, that do not conform to conventional or time frequency analyses. In the hierarchical model, the first level describes an observation model for multiple ERPs. The second level models the first-level parameters over subjects and trial types. These contain the differences or treatment effects one hopes to elicit by experimental design. Critically, to derive valid statistics at the second-level, one has to estimate the associated error covariance. It transpires that one can choose the observation model, at the first-level, to finesse nonsphericity estimation at the second. We will illustrate this using a discrete wavelet transform at the first level. The wavelet transform has two important features. The first is that it decomposes the ERP in the time frequency domain, which gives a sparse representation of its salient features. The second is that the wavelet transform affords an efficient error covariance estimation at the second level. However, we stress that other useful (linear) transforms like the Fourier transform can also be used in the two-level approach. The approach described here pertains to the analysis of single voxel data. We assume that the error covariance matrix, at the second level, is known. In a subsequent communication, we will describe the estimation procedure in a mass-univariate setting (i.e., to spatially reconstructed ERP data). This paper comprises three sections. In the first, we describe the mathematical basis of hierarchical models, paying special attention to the formulation of conventional analyses within this framework. In the second section, we provide worked examples using simulated and real data to demonstrate the use and flexibility of our approach. We conclude with a discussion of how this procedure relates to other analyses in the literature. Hierarchical models This section establishes the temporal model that is used, in various forms, in the next section. This is a two-level hierarchical Fig. 1. Example temporal matrices. (a) Full wavelet transform (Daubechies 4) in design matrix form. (b) Truncated wavelet matrix X t generated using the Daubechies 4 set. We removed the two highest scales from the full wavelet transform giving 64 wavelet regressors out of a potential 256. (c) First-level design matrix X (1) for two subjects with two conditions each.

3 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) model (where we estimate model parameters in a two-stage procedure, see below). The model is y ¼ X ð1þ b ð1þ þ ð1þ b ð1þ ¼ X ð2þ b ð2þ þ ð2þ where y is the data vector, X (1) and X (2) are design matrices, b (1) and b (2) are parameter vectors, and (1) and (2) are normally distributed error vectors. The data y comprises stacked ERPs y ij, where i =1,..., N subjects, j =1,...,N types. N subjects is the number of subjects and N types is the number of trial types. First-level model specification The model at the first level is described by the design matrix X ð1þ ¼ I NsubjectsN types X t and the error covariance matrix C ð1þ ¼ diagðk ð1þ ÞC t where X t is the N bins N p matrix that models the temporal components of a single ERP. The operator diag() returns a matrix with the vector argument on its main diagonal and zeros elsewhere. The observation error covariance matrix C (1) can be decomposed into ERP-specific components weighted by variance parameters k k (1), where k =1,...,N subjects N types (see below). The matrix X t can represent any linear transform of the ERPs. Here, we consider a wavelet transform. The wavelet transform enables the decomposition of the data into time frequency components. An excellent and intuitive introduction to the wavelet transform can be found in Gershenfeld (1998). A more rigorous treatment is provided by Strang and Nguyen (1996). In our implementation, we use an explicit matrix representation, which is a sequence of multiplications of filter and permutation matrices (Gershenfeld, 1998; Press et al., 1992). The matrix representation allows us to formulate the wavelet model as a design matrix. Moreover, as illustrated by the SPM package for the analysis of functional magnetic resonance imaging (fmri) data, the design matrix can be portrayed as an image to provide an intuitive understanding of the model. In this paper, we use the Daubechies wavelets of order 4 (Daub4) (Daubechies, 1992). This wavelet transform is one of the more commonly used in the literature. The Daub4 transform is an orthogonal wavelet transform with compact support. The implicit transform is framed in terms of parameter estimation within the hierarchical model. An example of the temporal matrix X t and an associated first-level design matrix is shown in Fig. 1. If X (1) is square, i.e., if we use a nontruncated wavelet transform, the covariance matrix of (1), C (1), cannot be estimated and Eq. (1) ceases to be an observation model. Otherwise, if N p < N bins, (1) is assumed to be normally distributed with (1) f N(0,C (1) ). The matrix C (1) can be decomposed into between-erp components (e.g., different levels of observation noise denoted by k k (1) ) and the within-erp component C t (Eq. (3)). For simplicity, we will assume homogeneous variance over peristimulus time and within subject with zero covariances; that is, the measurement error in each ERP is assumed to be white noise with ERP-specific variance so that ð1þ ð2þ ð3þ C t = I Nbins. This white-noise assumption can be motivated by assuming that the physiologically mediated within-erp noise is dominated by the measurement noise. In our model, random physiological effects reside in (2) (see below and Summary and discussion). This gives us components Q k t, k =1,...,N subjects N types (c.f. companion paper, Eqs. (3) and (4)). These are ordered in a block diagonal fashion so that C ð1þ ¼ X k k ð1þ k Q t k ð4þ ¼ diagðk ð1þ ÞI Nbins ¼ diagðk ð1þ ÞC t An example of first-level covariance components is shown in Fig. 2. The vector k (1) has length N subjects N types and consists of the first-level variance parameters k (1) = k (1) 1,...,k (1) N subjects N types. This means that the observation noise variance can be different for each subject and condition. Second-level model specification The second-level design matrix X (2) models the first-level parameters over subjects, X ð2þ ¼ X d I Np and C ð2þ ¼ XN d comp l XN p comp m k ð2þ lm ðqd l Q p m Þ where the experimental design matrix X d encodes subject- and trialtype-specific treatment effects. The second-level error covariance Fig. 2. Example of first-level covariance components Q k (1). We modelled four ERPs (from an experiment with two subjects with two trial types each). Each component models the variance of a single ERP. ð5þ ð6þ ð7þ ð8þ

4 506 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) N comp N comp (2). matrix C (2) is modelled using a linear mixture of covariance components. As for the design matrix X (2), C (2) can be decomposed into design-specific components Q d (e.g., within-subject correlations) and physiological error Q p. Although this (physiological) random effect is treated as error, from the point of view of the statistical model, it could well represent highly structured variation in event-related processes. The error sources are modelled using d p components. These components are weighted by variance parameters k lm Using X d to generate the second-level design matrix provides a general approach that can embrace many different experimental designs, e.g., modelling condition-specific effects or group comparisons. An example of X (2) based on X d =1 Nsubjects I Ntypes (a simple averaging matrix that averages over subjects across trial types) is shown in Fig. 3. (1 N denotes a column vector of ones of length N). The physiological error components Q p m are generally nonspherical, even with the reparameterisation implicit in wavelet transforms, because of intersubject variation in the way ERPs are expressed. A prominent feature of intersubject error is that its variance changes with peristimulus time. For example, at prestimulus time points, low error variances prevail, whereas the variance increases around phasic components like the N1 and then decreases again. This sort of nonsphericity is probably due to latency variations and other mechanisms that shape neuronal transients. Any model, for the second-level error, needs to capture this nonstationary variance and attending correlation structure. This is not a simple task, because one does not usually have enough observations to estimate a highly parameterised model of the error at a single voxel. There are two principled approaches to this covariance component estimation issue. The first is to use prior knowledge about the ERPs structure to reduce the number of firstlevel parameters and associated covariance parameters at the second level. The second is to compute precise empirical estimates using data from other voxels. In this paper, we assume that we know the structure of the error covariance matrix at the second level, C (2), and defer the description of its estimation for spatially extended ERP data, using the second approach, to a subsequent communication. Hierarchies and empirical Bayes An important aspect of hierarchical observation models is that they lend themselves to an empirical Bayesian interpretation. In the current model, the second-level error matrix C (2) is equivalent to the prior covariance of the parameters b (1) at the first level (Friston et al., 2002), i.e. C ð2þ ¼ Covð ð2þ Þ¼Covðb ð1þ Þ ð9þ This means C (2) can be regarded as describing intersubject variability, or as placing prior constraints on the subject-specific estimate of b (1). Although not pursued here, this perspective on C (2) leads to a parametric empirical Bayes approach in which C (2) can be used as a shrinkage prior on conditional estimates of b (1). See Friston et al. (2002) for details. In this paper, we are interested in estimating b (2) as opposed to b (1). Because there are no priors on the second-level parameters, conditional and maximum-likelihood estimates reduce to the same thing and the empirical Bayes perspective can be discounted. However, the notion of treating C (2) as a prior covariance has implications for the choice of X t.we will return to this in the discussion. Estimation After model specification (Eqs. (1) (8)), we estimate the parameters using maximum likelihood (ML) or ordinary least squares (OLS). Variance parameter estimates are obtained using restricted maximum likelihood (ReML). To render the method computationally efficient, a two-stage procedure can be used (Holmes and Friston, 1998): This entails (i) estimating the firstlevel parameters for each subject and trial type and taking the ensuing estimates bˆ (1) to the second level. (ii) At the second level, we compute ML or OLS estimates of b (2) and ReML estimates of k (2). We can do this in two distinct stages because X t (Eq. (2)) is the same for all subjects and trial types. For ML estimators at the second level, we would require k (1) k1 c k (1) k2 for any pair of ERPs k 1 and k 2 (Eq. (4)). When modelling inhomogeneous variances over subjects, at the first level, the two-stage procedure remains valid but the ensuing generalised least-squares estimators are no longer ML. However, we expect that the deviation from the ML estimates is small compared to the typically observed between-subject variability at the second level. In this paper, we use OLS estimates to enable a more direct comparison with conventional approaches. However, in practice, ML estimates are preferable, because they are more precise and lead to more sensitive inferences. Inference contrasts Fig. 3. Example of second-level design matrix X (2) for two subjects with two trial types. This design matrix models the trial type-specific averages. At the first level, each trial type is modelled using N p = 64 wavelet regressors. After parameter estimation, the next step is to make inferences about specific effects. In a classical setting, this proceeds using contrast vectors (or matrices) that specify null hypotheses (H 0 ). After parameter estimation, we derive a statistic that tests H 0. If the associated P value is small enough, one can reject H 0. In this section, we describe the generation and use of contrasts. In the next section, we will deal with statistical inference, i.e., the estimation of the P value associated with the contrast. An introduction to contrasts in the context of linear models is given in Poline et al. (2004). A contrast is a linear combination of parameter estimates that defines a specific null hypothesis about the parameters. Within our model, the parameters are wavelet

5 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) coefficients. However, it is not always intuitively obvious how null hypotheses can be specified in wavelet space. This problem can be solved by formulating contrast weights in peristimulus time or in the (peristimulus) time frequency domain and projecting them to the second level. Generation of second-level contrast vectors Because we are dealing with a hierarchical observation model, the contrast vector factorises into two terms c (2) = c d c w. The vector c d prescribes the comparison at the level of the experimental design (e.g., [ 11] T to compare the two trial types in the examples above). This component specifies which ERPs should be tested. The second component c w specifies what aspect of the ERP we are interested in. This is usually some ERP component that is restricted in time frequency. The simplest example would be a peristimulus time window defined by a vector w with w i = 1 within the window and w i = 0 elsewhere. The within-erp contrast weights c w are defined by w through c wt ¼ w X t c ð2þ ¼ c d c w ð10þ where w is the generalised inverse of w. For a simple time window, w y i is the average of the ith ERP in the window (Fig. 4). More generally, w will be a matrix spanning a subspace of X t that is restricted to a time frequency window. The contrast c (2)T bˆ (2) can then be tested in the usual way by forming t- or F-statistics (see next subsection). This projection procedure is general in the sense that it factorises the second-level Fig. 4. Creating contrasts at the second level, where the hypothesis is specified over two subjects with two ERPs each (window averages). Left: hypothesis in terms of design; right: hypothesis in terms of what ERP features exhibit the design effect.

6 508 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) Fig. 5. Example of a two-dimensional contrast component, see text for description. Left: hypothesis w in measurement space; right: contrast component c w in wavelet space. contrast vector c (2) into two components c d and c w. A change in the question at either the design level or at the level of the ERP can then be implemented by changing a single component. For example, more elaborate hypotheses like interactions in factorial designs can be specified by adjusting c d. To test other time frequency windows, one only needs to change c w. Note that, throughout the paper, our working assumption is that the basis set X t captures most of the interesting variability in ERP data. This is true for the basis sets and the data analysed in this paper. However, if X t misses too much of the experimentally induced variance, contrasts at the second level become difficult to interpret, because their projection back onto measurement space may not conform to what was required (i.e., w p X t X t w). Second-level contrast matrices Specifying hypotheses in this way can be generalised in a useful and powerful way by allowing w to be a N bins N con matrix. This leads to multidimensional second-level contrasts. For example, in fmri, contrast matrices are useful when testing for responses mediated by haemodynamic response functions (HRF) of unknown form. In this case, the haemodynamic response function is modelled by basis functions (the principal response shape and two first partial derivatives; Friston, 2002). One can then form a threecolumn contrast matrix that tests for any effect spanned by regressors formed from the three basis functions. For ERP data, the same principle can be used to test for effects that are modelled by multiple columns in w. A simple example, which builds upon the window averaging approach described above, is to test for effects distributed over multiple windows. In the simplest case, instead of testing for a single window, we can test for any effect in two or more neighbouring time windows. Fig. 5 shows two columns of the resulting matrix w. Importantly, this approach allows for a large range of tests in the time frequency domain. Contrast matrices are particularly useful for testing frequency-specific subspaces. Suppose we want to test for the expression of power at a given frequency in a particular peristimulus time window. This can be achieved by using a matrix w with two columns that comprises two modulated sinusoids with Fig. 6. Example of a two-dimensional time frequency w matrix. Top: matrix w in measurement space for a single ERP. Bottom: associated c w in wavelet space.

7 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) p/2 relative phase. The modulation can be any window function (e.g., a Gaussian) centered on the peristimulus time of interest. Such a time frequency contrast matrix is shown in Fig. 6. The notion of factorising hypothesis specification into a design component and an ERP component is at the heart of the generalisation of classical approaches proposed in this paper. Normally, in conventional analysis, the ERP component is completely predetermined by the window averaging applied to the data. In our application, this averaging represents one of many hypotheses that could have been tested, possibly jointly, and is specified through the second-level contrast. Operationally, this means the user has to specify the design contrast and the ERP contrast separately, the latter in observation space. A potentially useful and flexible way of specifying the ERP component is in terms of a time frequency window (see Fig. 7). This allows the user to test for any response that lies within certain peristimulus time and frequency bounds. A temporally long, frequency-restricted window would implicitly test for stimulus-locked transients in a prespecified frequency band. In contradistinction, a short window, covering multiple frequencies, would test for a temporally localised shape of unspecified form or frequency. Critically, in the limiting case that the window includes only the lowest frequency (i.e., constant term), the resulting contrast is onedimensional and conforms to the temporal window. This is exactly the same as the classical time-window averaging approach. In short, multidimensional contrasts admit hypotheses about time frequency effects that subsume the classical testing procedure. This is illustrated graphically in Fig. 7, which should be compared to Fig. 7. Creating time frequency contrast matrices at the second level, where the hypothesis is specified as a rectangle in the time frequency domain. Left: hypothesis in terms of design, i.e., the average over two trial types; right: hypothesis in the ERP time frequency domain. The hypothesis is whether there is any activity around time bin 50, in the frequency band between 1 and 28 Hz. Eight contrast vectors c w are left after orthogonalisation (see text). These are then used to generate the second-level contrast matrix c (2).

8 510 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) Fig. 4. Note that we use an orthogonalisation of the contrast matrix to remove (nearly) redundant contrast vectors. These can arise when using small, e.g., 1 Hz, steps to cover a frequency range in ERP time series that are sampled at a low rate, e.g., 200 Hz. This generalised approach relies upon multidimensional contrasts that are tested with the F statistic. The ability to relax assumptions about the shape of ERP differences at particular peristimulus times may improve sensitivity when these shaped differences do not conform to the simple box car averaging forms implicitly assumed by classical averaging techniques. The ensuing contrast component c w enters into the Kronecker tensor product to form the full second-level contrast c (2) in the usual way. Note also that one can use multidimensional design contrast matrices c d. This also gives second-level contrast matrices c (2), where multiple effects are tested over ERPs as opposed to within ERPs. For example, one can test for a response elicited by either of two trial types by using c d = I 2. Below, we will illustrate the general principle of single- and multidimensional contrasts using analyses of simulated and real data. Fig. 8. Summary of contrast generation in the hierarchical and conventional approach.

9 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) Fig. 9. Real and simulated ERPs. Left: 16 randomly chosen ERPs out of 54 at channel PO8. Right: 16 simulated ERPs generated by using a covariance matrix based on real data (see text). Inference statistics Having specified second-level contrasts, we can make inferences using the estimated parameters. Note that a second-level contrast specifies a random effects analysis. In this subsection, we describe the formation of t and F statistics to test null hypotheses about these contrasts. The t statistic is given by t ¼ cð2þt ˆbð2Þ ˆ Stdðc ð2þt ˆbð2Þ Þ ð11þ where c (2) is a second-level contrast vector and Std ˆ denotes the estimated standard deviation. The parameters can be estimated using generalised least squares to give ML estimators. Their standard deviation is estimated in the usual way using ReML variance component estimators (Friston et al., 2002). However, because this paper uses OLS estimators, we are obliged to use a Satterthwaite approximation to compute effective degrees of freedom (Worsley and Friston, 1995; Kiebel et al., 2003). P values can then be computed using the cumulative density function (CDF) of the null distribution of the sampled t value. When using a contrast matrix (or vector), one implicitly defines a reduced model. In the context of nested models, the F statistic is used to compare the corresponding sum of squares: F ¼ T ð1þ ˆb M ˆbð1Þ m 2 ˆb ð1þt R ˆb ff m1;m ð1þ m 2 1 ð12þ where M = R 0 R, R = I X (2) X (2) is the residual forming matrix of the full model. Matrix R 0 is the residual forming matrix of the reduced model, where R 0 = X (2) c (2) 0 and c (2) 0 = I c (2) c (2) I. The degrees of freedom m 1 and m 2 are estimated using a Satterthwaite approximation. P values can be derived from the CDF of the null distribution of F. 1 In summary, this section has described how to formulate a twolevel hierarchical model with multiple covariance components, estimate model parameters, specify contrasts and estimate P values. In the following, we will deal with the conventional model for ERP data. This model is a special, if trivial, case of the hierarchical two-level model. We will then compare conventional and hierarchical analysis procedures using simulated and real ERP data. The conventional model Fig. 10. Comparison of P values on null data using a conventional and hierarchical model. The inter-erp variability of the synthetic data was derived from real data (see text). Results are displayed on a log log scale. Dashed line: P values required for an exact test, Dotted line: conventional analysis. Solid line: hierarchical model. In classical ERP research, one assumes that each single trial comprises the ERP and some (usually white) normally distributed measurement noise. Under these assumptions, a natural estimator for the ERP of subject i and trial type j, y ij, is the average of all single trials y ijn, n =1,...,N trials, i.e., ŷ ij =1/N trials S 1 Ntrials y ijn. Given ERP estimates for each subject i and each trial type j, one then computes contrasts w y ij, e.g., the average over a peristimulus time window. These contrasts then enter an analysis of variance (ANOVA). The ensuing nonsphericity, i.e., inequality of variances 1 The square of the t statistic is an F statistic. Therefore, in SPM, the t statistic provides for one-tailed tests only.

10 512 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) of each trial type or covariance between them, is usually accommodated using a Greenhouse Geisser correction. This correction approximates the null distribution of the F statistic by adjusting the effective degrees of freedom. An example of such an analysis can be found in Henson et al. (2003) and is the quasi standard in the experimental ERP literature. It is useful to formulate the conventional approach in terms of the two-level model. We can implement averaging over a peristimulus time window for all subjects and trial types as a contrast matrix c (1) = I Nsubjects N types w T. As above, w i = 1 for peristimulus times in the window and w i = 0 at all other times. The implicit model can be expressed mathematically as y ¼ X ð1þ b ð1þ ¼ b ð1þ c ð1þt b ð1þ ¼ X ð2þ b ð2þ þ ð2þ ð13þ Fig. 12. Comparison of P values on synthetic data using a conventional and two-level model. The data was generated as described in the text. The data contained ERP differences (an N170 component as shown in Fig. 11), which we tested for. Dashed line: P values under null hypothesis. Dotted line: conventional model. Solid line: two-level model. where X (1) is the identity matrix and X (2) = X d I 1 = X d.as above (Eq. (7)) the matrix X d encodes experimental design variables. The amplitude difference between trial types is assessed using a second-level contrast c (2) = c d =[ 1 1] T. Note here that there is no partitioning of the response into the observation noise and the physiological ERP variability. This means the conventional model assumes (1) = 0 and the two-level model reduces to a onelevel model. This is the first key difference between the conventional and the hierarchical model. The second is that the conventional model only passes a single contrast per ERP to the second level. This means that, in the conventional model, each hypothesis test is based on a separate model specification and estimation procedure. In contrast, the hierarchical model parameters are only estimated once. A family of hypotheses can then be tested at the second level without model refitting. The conventional approach precludes this because only contrasts of first-level parameters are taken to the second level. As will be illustrated in the next section, the two-level model enables us to test hypotheses that could never be tested in a conventional analysis. Furthermore, the conventional approach precludes hypothesis tests for treatment effects that span several dimensions (e.g., time frequency analyses). The fundamental differences between the hierarchical and nonhierarchical (i.e., conventional) procedures are summarised in Fig. 8. Illustrative analyses In this section, we apply the hierarchical and conventional methods to synthetic and real data. The synthetic data were designed to show that the hierarchical approach gives valid tests and retains sensitivity. Furthermore, we will demonstrate contrasts that can only be used in a hierarchical context. Analyses of real data are provided to illustrate the operational details, particularly contrast specification. Synthetic data Fig. 11. Signal component used for simulations. This component is the fourth eigenvector of the sample data covariance matrix of 54 ERPs. Synthetic data were constructed by sampling ERPs from a normal distribution whose moments were based on real ERP data. We selected a channel, PO8, which showed a N170 component in response to face stimuli (Henson et al., 2003). We used 18 subjects with three trial types giving 54 ERPs of length 256 sampled at 200 Hz. We computed the grand mean ERP and the singular value decomposition (SVD) of the sample residual covariance matrix. ERP data were simulated by drawing from the empirically defined multivariate Gaussian distribution with the sampled ERP mean. The covariance was computed using only the first 15 eigenvectors of the sampled covariance matrix. This corresponds to components Q p in Eq. (8). The restriction to the principal components of ERP variability biased the nonspherical variation, in simulated data, towards physiological as opposed to measurement sources of variance. In Fig. 9, we compare 16 randomly selected ERPs with

11 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) Fig. 13. Time frequency two-column contrast matrix to test for evoked power, at 40 Hz, averaged over two trial types. The modulating Gaussian is centered at stimulus onset and has a FWHM of 75 ms. 16 simulated ERPs. One can see clearly that the simulated data show many features of real ERPs. More importantly, the residual covariance structure of the simulated data is roughly the same as of real ERPs (not shown). In the following, we use these data for assessing the specificity and sensitivity of our approach in relation to conventional analyses. Each synthetic epoch consisted of 256 time points. Each simulation comprised six subjects with two trial types each giving 12 single-channel ERPs. For each set of simulations, we generated 10 4 data sets. In the first two simulations, we assess specificity and establish that the hierarchical and conventional approaches are valid. In the second set of simulations, we show that our approach is sensitive to a differential signal typically tested for by conventional analyses. In the third set of simulations, we use multidimensional contrasts to test for evoked power, in a time frequency window, over two trial types. In the final set of simulations, we illustrate how one can use a two-dimensional contrast to test for biphasic signals. Specificity In the first set of simulations, we generated synthetic data as described above. We tested for a difference in ERP amplitude over ms between the two trial types, over six subjects (note that this difference has zero expectation in the simulated data, i.e., there is no differential ERP signal). This peristimulus time window contains the N170 component (see also Fig. 17). As described above, the N170 hypothesis can be specified in measurement space by a vector w = 1 within the peristimulus time window and w =0at other times. For the conventional analysis, we computed 12 ERP-specific first-level contrasts c (1) Ty, where c (1) = I 12 w. The second-level design matrix X (2) = X d =1 Nsubjects I 2 implemented a simple averaging over subjects enabling a two-sample t test. We tested for an amplitude difference between the trial types using the secondlevel contrast vector [ 1 1] T. For the hierarchical approach, we used, at the first level, a full (eight scales) wavelet decomposition (Daubechies 4) and removed the two highest scales. This implements a truncation based on the prior assumption that the signal subspace does not span the highest two scales. All first-level parameter estimates bˆ(1) were brought to the second level. The second-level contrast vector was [ 11] T c w (Eq. (10)). We modelled the observation error (Eq. (4)) as homogeneous over the six subjects. At the second level, we assumed the error covariance matrix to be known. 2 The results of the simulations are shown in Fig. 10 as a P P plot. In this plot, lines above the identity represent invalid, or capricious performance, regions below the identity represent conservative performance. Both methods returned P values that are very close to the distribution necessary for an exact and valid test. Sensitivity In the second set of simulations, we generated synthetic data using the procedure described above. However, we added a differential signal to the ERPs of the second trial type. This signal conformed to a N170 component. The shape of the signal was taken from the fourth eigenvector of the sample data covariance matrix. This eigenvector captured the form of the N170 component (Fig. 11). In all simulations, we added this signal to the simulated ERPs of one trial type so that we generated a differential signal 2 We used a precise estimate based on data from all channels.

12 514 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) Contrast matrices and time frequency contrasts Fig. 14. Distribution of P values for null data testing for power at 40 Hz averaged over two trial types. The data was generated as described in the text. Results are displayed on a log log plot. Dashed line: P values required for an exact test. Solid line: two-level model. between trial types. The amplitude of this signal was 40% of the estimated average amplitude of the N170. We used the same model and contrasts as in the first set of simulations; that is, we tested for a difference in amplitude between trial types in the ms time window. In Fig. 12, the P P plot shows that both approaches give roughly the same sensitivity to this signal. The hierarchical approach seems to result in slightly more sensitive results, but this might be due to the Satterthwaite approximation for a single variance parameter (Kiebel et al., 2003). In these simulations, we illustrate hypothesis testing in the time frequency domain. We asked whether the power, of evoked oscillations at 40 Hz, averaged over two trial types, in a given peristimulus time window is greater than chance expectation. To generate second-level contrasts, we specified matrix w (Eq. (10)) as two modulated sinusoids in measurement space (cf. Fig. 6). Both sinusoids had a frequency of 40 Hz, where one was phase shifted by p/2. The sinusoids were windowed with a Gaussian centered at 0 ms in peristimulus time, i.e. at stimulus onset, with a full width at half maximum (FWHM) of 75 ms. This hypothesis was chosen, because we wish to assess the specificity of the resulting test (see below). To average over trial types, we used c d =[11] T (Eq. (10)). The ensuing columns of c (2) = c d c w are shown in Fig. 13. This way of testing for evoked oscillations is quite general and can be extended to test not only a specific frequency, but frequency ranges by using pairs of windowed sinusoids to cover the frequencies required (Fig. 7). We used simulated data to assess the specificity of inferences using the F statistic (Eq. (12)). To ensure the data conformed to the null hypothesis, we sampled simulated data, for this set of simulations only, from a multivariate normal distribution with zero mean. The results are shown in Fig. 14 and indicate valid and exact tests. This example demonstrates the utility of the two-level model because this analysis is precluded by the conventional approach (i.e., there is no single contrast vector c (1) that can be used to test for evoked oscillations of unknown phase). In this subsection we have introduced the use of multidimensional contrasts to test for evoked frequency-specific oscillations at a particular peristimulus time. In applying this contrast to null data (no average ERP difference), we hope to have established its validity by showing the false-positive rate conforms to its nominal value. In a later section, we will use a similar contrast to test a time frequency hypothesis in real data that did evidence significant power in the alpha band. Fig. 15. Contrast generating vectors in measurement space. These vectors are used to test for a difference between trial types in either a component around 100 or 170 ms.

13 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) Fig. 16. Distribution of P values when testing for a biphasic signal with the conventional and the two-level model. With the conventional model, we used three different contrasts to show that one cannot completely capture the biphasic signal with a single contrast vector. Dashed line: P values under the null hypothesis. Green line: two-level model. The other lines show the P values from the conventional model testing for a change in component 1 (red), in component 2 (blue), and a change in their average (magenta). Other multidimensional contrasts In this last set of simulations, we illustrate another application that is precluded by the conventional approach. We test for the difference, in a compound signal, between two trial types. The signal of interest consists of two components, the first (early) centered around 100 ms, the second (late) around 170 ms. The difference (between trial types) could be expressed in either or both components. In this case, a single contrast vector will not be optimally sensitive, because we have no prior knowledge about the relative contribution of the two component differences. Differences of unknown form like these can be assessed using a two-column contrast matrix c w. Each column supports one of the components in peristimulus time. In measurement space, we can specify each component in terms of a standard averaging window, or some other user-specified shape. Here, we chose a Gaussian form for w (Fig. 15). The test for any difference in the amplitudes of the Gaussians is implemented by defining c d =[ 11] T and computing the second-level contrast c (2) = c d c w. This principle can be generalised to more components by adding more Gaussians to the matrix w. The F statistic provides an overall test for any difference expressed over the components of w. The nature and form of any significant difference is characterised by the contrast of parameter estimates. We generated synthetic data using the procedure described above. We added differential signal that consisted of a biphasic signal (a mixture of two Gaussians). The amplitudes of the two Gaussians were drawn independently from a Gaussian distribution with zero mean and variance 4. The signal to noise ratio (SNR) is 0.47 (c.f. Fig. 17). In Fig. 16, we show the results of the simulations. For each of the 10 4 data sets, we tested the hypothesis of any difference, in either component, using the F statistic (Eq. (12)). For the conventional approach, we used three different single-dimensional contrasts for a two-tailed test based on the F statistic. Two of these tested for a difference in a single component, and the third tested for a mean difference over both components. These tests show the sensitivity one can hope for with the conventional approach. The hierarchical model with a two-dimensional contrast leads to a test that is at least as sensitive as the most sensitive test based on the conventional model. This is because the alternative hypothesis spans both dimensions of the real treatment effect. Applications to real ERP data Here, we describe an analysis of (real) ERP time series using the conventional and hierarchical models. The hypotheses tested below were chosen because they illustrate many of the procedural details and concepts of the two-level approach. The data were acquired during a memory study, which involved the presentation of faces and scrambled faces to 18 subjects (Henson et al., 2003). Fig. 17. Preprocessed single-channel ERP data as a function of peristimulus time. Blue: ERPs of trial type 1 (unfamiliar faces); red: ERPs of trial type 2 (scrambled faces). Fig. 19. Time frequency representation of the averaged power of multisubject ERP data at channel PO8 between 6 and 20 Hz over peristimulus time.

14 516 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) Fig. 18. Fitted back-projected data, in measurement space, from the conventional and hierarchical model. (Left) Conventional model; (right) two-level model. Solid lines: trial type 1; dashed line: trial type 2. The fitted data from the two-level model shows a jagged shape because of the truncated wavelet transform. An ERP analysis The EEG was recorded from 29 silver/silver chloride electrodes using an elasticised cap (Falk Minow Easycap montage 10, plus an electrode on each mastoid. Recordings were made with reference to a mid-frontal electrode and algebraically re-referenced. Impedances were nearly always less than 5 K V. Vertical and horizontal electro-oculograms (EOG) were recorded from electrode pairs situated above and below the right eye and on the outer canthi. EEG and EOG were amplified with a bandwidth of Hz (3 db points) and digitised (12 bit) at 200 Hz. The recording epochs began 100 ms before stimulus onset (baseline) and lasted 1280 ms. The data were preprocessed using SPM2 and ERP-specific extensions. All preprocessing functions are implemented either as Matlab (Matlab 6.5, The MathWorks) or C routines. Trials that contained blinks, horizontal or nonblink eye movements, A/D saturation, or EEG drifts were rejected from visual inspection without knowledge of trial types. Trials were averaged according to two trial types. The first comprised unfamiliar faces, the second scrambled faces. See Henson et al. (2003) for a detailed description of the experimental design. All ERP waveforms were based on a minimum of 60% artefact-free trials per trial type (approximately 25 on average). The average waveforms were low-pass-filtered to 20.7 Hz using a zero-phase-shift filter. Each of the 36 ERPs (two trial types per subject) had 256 data points. From these ERP data, we selected one channel, Easycap site 41, corresponding approximately to PO8 in the extended system. This channel was shown in Henson et al. (2003) to demonstrate a significant differential response between trial types. The preprocessed single channel data are shown in Fig. 17. We were interested in a difference between trial types around 170 ms in peristimulus time. A difference in ERP amplitude around this time indicates a difference in the N170 component (Bentin and Golland, 2002). The expression of this component can be tested with a contrast that expresses the amplitude difference in a window from 150 to 190 ms. The corresponding within-erp component w was generated as illustrated in Fig. 4. The second-level contrast vector c (2) was obtained by computing c d c w (Fig. 8). We used the Daubechies 4 wavelet transform with eight scales, with truncation of the three highest scales. This gave five scales with 32 wavelet parameters per trial type, i.e., 64 parameters at the second level. Note that the band-pass filter (see above) applied to the data should be, strictly speaking, incorporated into the model. This could be done with premultiplication of the first-level equation with a filter matrix or, alternatively, by adding some discrete cosine or Fourier transform regressors to the temporal design matrix X t. After OLS parameter and ReML variance parameter estimation, we computed the t value (Eq. (11)) and effective degrees of freedom as t = 3.80 and m = The corresponding P value was P = Fig. 20. Images of the contrast components used for testing the power at 10 Hz around 170 ms. Left: two-column matrix w. Right: corresponding second-level contrast matrix c (2) with c d =[11] T.

15 S.J. Kiebel, K.J. Friston / NeuroImage 22 (2004) For the conventional analysis, we used the contrast matrix c (1) = I 36 w. We analysed the resulting 36 contrasts, at the second level, using a two sample t test (X d =1 18 I 2 and c (2) =[ 1 1] T ). This gave a t statistic (t = 3.59) with m = 27.99, where we allowed for unequal variances and between trial-type covariance. The corresponding P value was The fitted trial typespecific responses, for both models, are shown in Fig. 18 after projecting the second-level parameter estimates, averaged over subjects, back onto peristimulus time. One can see clearly the differential response. The fitted data from the two-level model shows a jagged appearance, especially around high-frequency peaks like the N170. This is due to the truncation of the wavelet transform from 256 to 32 parameters. A time frequency analysis The second hypothesis we tested pertained to power evoked at a specific frequency, in a particular peristimulus time window. We tested whether there was greater power, at 10 Hz, averaged over the two conditions, in a poststimulus time window centered on 170 ms. 3 This hypothesis was tested using contrasts based on two modulated sinusoids in measurement space. To illustrate the time frequency structure of the data, we estimated ERP power, averaged over both trial types, using the continuous Morlet wavelet transform (Fig. 19). The data consisted of 36 ERPs. This shows clearly the excess of alpha power around ms. Note that the Morlet transform is a continuous wavelet transform. This means we cannot use it directly in the present framework, because it has more wavelet coefficients than data points. However, it is well-suited to describe the time frequency structure of time series (Kronland-Martinet et al., 1987; Tallon-Baudry and Bertrand, 1999). The Morlet wavelet consists of two sinusoids that are modulated by a Gaussian window. The wavelet at frequency f 0 is defined as w(t) = exp ( t 2 /(2r 2 t)) exp(2ipf 0 t), with r t = z 0 /(2pf 0 ); that is, the userspecified factor z 0 fixes the ratio between the temporal variance of the Gaussian and the frequency f 0. We used the same two-level model (and its parameter estimates) as in the previous subsection. The modulated sinusoids (see the simulated data section) were projected to form the second-level contrast matrix c (2) (Eq. (10) and Fig. 20). We tested the hypothesis of evoked alpha, at 170 ms, using the F statistic (Eq. (12)). The resulting F value was 7.13 with effective degrees of freedom of 1.31 and 49.07, giving a P value of and rendering the effect significant. Note that we cannot test this hypothesis in a conventional ERP framework. Summary and discussion We have described a temporal model adopted by SPM for ERP data. The model pertains to voxel/channel data. To analyse time series of source-reconstructed ERP images, one needs a spatial model, which will be described in a future communication. The methods described here are implemented in Matlab software compatible with the SPM2 distribution and will be an integral part of future SPM releases. 3 With the two-level model, such a test is about stimulus-locked oscillations in evoked responses. Induced oscillations require a slightly different approach (see Summary and discussion). Two-level models Our temporal model conforms to a two-level hierarchical linear model with normally distributed error. The response variable consists of concatenated ERP data for one location. At the first level, the design matrix is based on an orthogonal wavelet set. The error at the first level (observation error) is assumed to be white. At the second level, we model changes in wavelet parameters caused by design and nonsphericity induced by between-subject variability in the expression of ERPs and experimental design. In the examples used to demonstrate the approach, the parameters were estimated using OLS and ReML in a two-stage procedure. Inference proceeds using second-level contrasts, corresponding to random effects analyses. This scheme can be adapted to ML and ReML estimates using weighted least-squares as described elsewhere (Friston et al., 2002). We used OLS parameter estimates to make the comparison with classical procedures more direct. Statistics that use OLS estimates entail an adjustment to the degrees of freedom. The usual choice for this adjustment is based on the Satterthwaite approximation (c.f. Greenhouse Geisser), which we also used for the estimation of null distributions. Had we used ML estimates, this adjustment would not have been necessary. Conventional contrasts The two-level model was validated using synthetic data. For real data and conventional hypotheses (like differences in time window averages), we found that one obtains roughly the same sensitivity as in a conventional analysis. This is because both approaches have to estimate the same thing, the variance of the tested contrasts. The conventional approach simplifies the nonsphericity structure of the second-level error terms because only one physiological measurement [contrast] from each ERP enters the second level. In the two-level model, one solves the same problem in two steps. First, one models and estimates the nonsphericity of the second-level error covariance matrix. Second, one forms a contrast of second-level parameters and estimates its variance by reference to the estimated nonsphericity. Ideally, both approaches should lead to the same specificity and our simulations showed that to be the case. Is the two-level approach better than the conventional approach? The conventional approach pays a price for reducing the nonsphericity structure to a single number. The first disadvantage is that only one contrast per ERP is modelled at the second level. Any inferences about responses that span multiple contrast vectors, within-erp, are precluded. As shown above, this limitation can be severe. For example, time frequency hypotheses cannot be tested. In contradistinction, the two-level model subsumes not only conventional ERP analyses, but also enables simple time frequency analyses. This is achieved by allowing for multidimensional contrast components c w (Eq. (10)) so that one can test for treatment effects that encompass more than one dimension. Inference about these effects are made using the F statistic. Other inferences that are precluded by the conventional approach comprise tests for response components whose form is not known. We illustrated this by testing for compound signal differences. The use of the contrast matrix enlarges the range of tests that are available. In short, the two-level model embraces not only the conventional analysis, but